每週問題 April 3, 2017

找出所有的可交換矩陣。

Find all matrices commuting with E, where E is the matrix all elements of which are equal to 1.

 
參考解答:

A=[a_{ij}]n\times n 階矩陣,r_i=\sum_{k=1}^na_{ik}c_j=\sum_{l=1}^na_{lj}1\le i,j\le n。因此,

EA=\begin{bmatrix} c_1&\cdots&c_n\\ \vdots&\ddots&\vdots\\ c_1&\cdots&c_n \end{bmatrix},~~AE=\begin{bmatrix} r_1&\cdots&r_1\\ \vdots&\ddots&\vdots\\ r_n&\cdots&r_n \end{bmatrix}

比較上式可知 EA=AE 等價於 c_1=\cdots=c_n=r_1=\cdots=r_n

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